OK, so no luck at all with documentation beyond the readily available operator's manual. I'm awaiting a reply from X-Rite though.
I did have some communications with Sam Coniglio (an authorized service provider) who indicated he had binned his 310 documentation "long ago". He did tell me to check the lithium battery as it likely needed replacing.
So off comes the skin... easy access from the top once the screws holding the plexi top panel are removed. The panel and the whole top half of the unit just pop off then leaving the main display circuit boards accessible. Lo and behold, right there on the back of the right hand board is a big-ass lithium battery. It appears said battery has been replaced at least once in the past as witnessed by the non-factory solder job on it's leads.
So a new battery has been ordered (pricing ranging from just under $5 to over $20 for what appears to be the same battery). And a number of the US suppliers will not ship it to Canada, nor list it as available on the Canadian versions of their websites, though a number of suppliers in do exist here in Canada.
I've included a couple of images of the insides of this beast. Wow, they don't make too many things like this anymore. It's a hugely solid aluminum casting, no flimsy showing here. The circuit boards actually contain components I recognize from my time in the electronics world. Solid late '80s devices here. Most of them big enough to actually solder by hand if necessary. No SMD.